Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

A Theory of Sampling for Continuous-time Metric Temporal Logic

This paper revisits the classical notion of sampling in the setting of real-time temporal logics for the modeling and analysis of systems. The relationship between the satisfiability of Metric Temporal Logic (MTL) formulas over continuous-time models and over discrete-time models is studied. It is shown to what extent discrete-time sequences obtained by sampling continuous-time signals capture the semantics of MTL formulas over the two time domains. The main results apply to "flat" formulas that do not nest temporal operators and can be applied to the problem of reducing the verification problem for MTL over continuous-time models to the same problem over discrete-time, resulting in an automated partial practically-efficient discretization technique.Comment: Revised version, 43 pages

An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part II)

There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics. In this Part II, we deal with the cases of all dense and the case of all unbounded linearly ordered sets.Comment: This is Part II of the paper `An Integrated First-Order Theory of Points and Intervals over Linear Orders' arXiv:1805.08425v2. Therefore the introduction, preliminaries and conclusions of the two papers are the same. This version implements a few minor corrections and an update to the affiliation of the second autho

Minimal Paradefinite Logics for Reasoning with Incompleteness and Inconsistency

Paradefinite (`beyond the definite\u27) logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for defining paradefinite logics, consisting of four-valued matrices, and study the better accepted logics that are induced by these matrices

Integrated Modeling and Verification of Real-Time Systems through Multiple Paradigms

Complex systems typically have many different parts and facets, with different characteristics. In a multi-paradigm approach to modeling, formalisms with different natures are used in combination to describe complementary parts and aspects of the system. This can have a beneficial impact on the modeling activity, as different paradigms an be better suited to describe different aspects of the system. While each paradigm provides a different view on the many facets of the system, it is of paramount importance that a coherent comprehensive model emerges from the combination of the various partial descriptions. In this paper we present a technique to model different aspects of the same system with different formalisms, while keeping the various models tightly integrated with one another. In addition, our approach leverages the flexibility provided by a bounded satisfiability checker to encode the verification problem of the integrated model in the propositional satisfiability (SAT) problem; this allows users to carry out formal verification activities both on the whole model and on parts thereof. The effectiveness of the approach is illustrated through the example of a monitoring system.Comment: 27 page

Quantified Propositional Gödel Logics

It is shown that Gqp↑, the quantified propositional Gödel logic based on the truth-value set V↑ = {1 - 1/n : n≥1}∪{1}, is decidable. This result is obtained by reduction to Büchi's theory S1S. An alternative proof based on elimination of quantifiers is also given, which yields both an axiomatization and a characterization of Gqp↑ as the intersection of all finite-valued quantified propositional Gödel logics

An integrated first-order theory of points and intervals : expressive power in the class of all linear orders

There are two natural and well-studied approaches to temporal ontology and reasoning, that is, pointbased and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. Recently, a two-sorted point-interval temporal logic in a modal framework in which time instants (points) and time periods (intervals) are considered on a par has been presented. We consider here two-sorted first-order languages, interpreted in the class of all linear orders, based on the same principle, with relations between points, between intervals, and intersort. First, for those languages containing only interval-interval, and only inter-sort relations we give complete classifications of their sub-fragments in terms of relative expressive power, determining how many, and which, are the different two-sorted first-order languages with one or more such relations. Then, we consider the full two-sorted first-order logic with all the above mentioned relations, restricting ourselves to identify all expressively complete fragments and all maximal expressively incomplete fragments, and posing the basis for a forthcoming complete classification